let 7*x be the least multiple where x is integer.

7x + 1 when divided by 2, 3, 4, 5, and 6 leaves reminder 0

thus

the smallest number divisible by 2,3,4,5 and 6 is the least common multiple of 2,3,4,5 and 6 that is 30

that is we can write

7x + 1 = 30k, for some k integer

lets find k

for k = 1 => 7x = 29 is not the solution

for k = 2=> 7x = 59 is not the solution

for k = 3 => 7x = 89 is not the solution

for k = 4 => 7x = 119 is the solutions since 119 : 7 = 17 and no reminder.

the least multiple of 7 is 119.